Definition:Constructed Semantics/Instance 1

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Let $\mathcal L_0$ be the language of propositional logic.

The constructed semantics $\mathscr C_1$ for $\mathcal L_0$ is used for the following results:


Define the structures of $\mathscr C_1$ as mappings $v$ by the Principle of Recursive Definition, as follows.

Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$.

Let a mapping $v: \mathcal P_0 \to \{ 1, 2 \}$ be given.

Next, regard the following as definitional abbreviations:

\((1)\)   $:$   Conjunction       \(\displaystyle \mathbf A \land \mathbf B \)   \(\displaystyle =_{\text{def} } \)   \(\displaystyle \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) \)             
\((2)\)   $:$   Conditional       \(\displaystyle \mathbf A \implies \mathbf B \)   \(\displaystyle =_{\text{def} } \)   \(\displaystyle \neg \mathbf A \lor \mathbf B \)             
\((3)\)   $:$   Biconditional       \(\displaystyle \mathbf A \iff \mathbf B \)   \(\displaystyle =_{\text{def} } \)   \(\displaystyle \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) \)             

It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by:

\(\ds v \left({\neg \phi }\right)\) \(:=\) \(\ds \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 1$}\end{cases}\)
\(\ds v \left({ \phi \lor \psi }\right)\) \(:=\) \(\ds \begin{cases} 1 & : \text{if $v \left({\phi}\right) = v \left({\psi}\right) = 1$} \\ 2 & : \text{otherwise}\end{cases}\)


Define validity in $\mathscr C_1$ by declaring:

$\models_{\mathscr C_1} \phi$ if and only if $v \left({\phi}\right) = 2$